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documentation for a sale of the valuable manuscript. Stored secretly for years by
Sirieix in his cellar, the palimpsest suffered damage from water and mold. In addition, after its disappearance from the Greek Orthodox Patriarchate's library, a forger added copies of medieval evangelical portraits in gold leaf onto four pages in the book in order to increase its sales value, further damaging the text. These forged gold leaf portraits nearly obliterated the text underneath them, and x-ray fluorescence imaging at the
1816:
382:, Dr. William A. Christens-Barry from Equipoise Imaging, and Dr. Keith Knox (then with Boeing LTS, now retired from the USAF Research Laboratory) used computer processing of digital images from various spectral bands, including ultraviolet, visible, and infrared wavelengths to reveal most of the underlying text, including of Archimedes. After imaging and digitally processing the entire palimpsest in three spectral bands prior to 2006, in 2007 they reimaged the entire palimpsest in 12 spectral bands, plus
164:
270:, the world's authority on Archimedes, realized that the work was by Archimedes. When Heiberg studied the palimpsest in Constantinople in 1906, he confirmed that the palimpsest included works by Archimedes thought to have been lost. Heiberg was permitted by the Greek Orthodox Church to take careful photographs of the palimpsest's pages, and from these he produced transcriptions, published between 1910 and 1915 in a complete works of Archimedes. Shortly thereafter Archimedes'
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130:. A Western businessman concealed the book for over 70 years, and at some point forged pictures were painted on top of some of the text to increase resale value. Unable to sell the book privately, in 1998 the businessman's daughter risked a public auction in New York contested by the Greek church; the U.S. court ruled for the auction, and the manuscript was purchased by an anonymous buyer (rumored to be
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difficult, while play could be awkward if pieces with sharp points are turned over. For the Codex board (again as with
Tangram) there are three ways to pack the pieces: as two unit squares side by side; as two unit squares one on top of the other; and as a single square of side the square root of two. But the key to these packings is forming isosceles right triangles, just as
887:– Socrates was arguing for knowledge by recollection, and here pattern recognition and memory seem more pertinent than a count of solutions. The Codex board can be found as an extension of Socrates' argument in a seven-by-seven-square grid, suggesting an iterative construction of the side-diameter numbers that give rational approximations to the square root of two.
658:, and balanced each line, or slice, of one figure against a corresponding slice of the second figure on a lever. The essential point is that the two figures are oriented differently so that the corresponding slices are at different distances from the fulcrum, and the condition that the slices balance is not the same as the condition that the figures are equal.
126:. Heiberg visited the church library and was allowed to make detailed photographs in 1906. Most of the original text was still visible, and Heiberg published it in 1915. In 1922 the manuscript went missing in the midst of the evacuation of the Greek Orthodox library in Istanbul, during a tumultuous period following
118:, at least). The complex manuscript was not appreciated at this remote monastery and was soon overwritten (1229) with a religious text. In 1899, nine hundred years after it was written, the manuscript was still in the possession of the Greek church, and back in Istanbul, where it was catalogued by the Greek scholar
386:: UV: 365 nanometers; Visible Light: 445, 470, 505, 530, 570, 617, and 625 nm; Infrared: 700, 735, and 870 nm; and Raking Light: 910 and 470 nm. The team digitally processed these images to reveal more of the underlying text with pseudocolor. They also digitized the original Heiberg images. Dr.
737:
Archimedes did not know about differentiation, so he could not calculate any integrals other than those that came from center-of-mass considerations, by symmetry. While he had a notion of linearity, to find the volume of a sphere he had to balance two figures at the same time; he never determined how
424:
is created when electrons traveling near the speed of light take a curved path around a storage ring—emitting electromagnetic light in X-ray through infrared wavelengths. The resulting light beam has characteristics that make it ideal for revealing the intricate architecture and utility of many kinds
661:
Once he shows that each slice of one figure balances each slice of the other figure, he concludes that the two figures balance each other. But the center of mass of one figure is known, and the total mass can be placed at this center and it still balances. The second figure has an unknown mass, but
622:
was a quantity Q approximated by two sequences, one providing an upper bound and the other a lower bound. If one finds two sequences U and L, and U is always bigger than Q, and L always smaller than Q, and if the two sequences eventually came closer together than any prespecified amount, then Q is
304:
Sometime between 1923 and 1930 the palimpsest was acquired by Marie Louis
Sirieix, a "businessman and traveler to the Orient who lived in Paris." Though Sirieix claimed to have bought the manuscript from a monk, who would not in any case have had the authority to sell it, Sirieix had no receipt or
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The fragmentary state of the palimpsest leaves much in doubt. But it would certainly add to the mystery had
Archimedes used the Suter board in preference to the Codex board. However, if Netz is right, this may have been the most sophisticated work in the field of combinatorics in Greek antiquity.
846:
in translating an unpointed Arabic text in which twice and equals are easily confused; Suter makes at least a typographical error at the crucial point, equating the lengths of a side and diagonal, in which case the board cannot be a rectangle. But, as the diagonals of a square intersect at right
490:
exhibit "Lost and Found: The
Secrets of Archimedes". In 2015, in an experiment into the preservation of digital data, Swiss scientists encoded text from the Archimedes Palimpsest into DNA. Thanks to its deciphering, some mathematicians suggest it is possible that Archimedes may have invented
874:
reveals that the number of ways to place the pieces of the Suter board to reform their square, allowing them to be turned over, is 17,152; the number is considerably smaller – 64 – if pieces are not allowed to be turned over. The sharpness of some angles in the Suter board makes fabrication
630:
Archimedes used exhaustion to prove his theorems. This involved approximating the figure whose area he wanted to compute into sections of known area, which provide upper and lower bounds for the area of the figure. He then proved that the two bounds become equal when the subdivision becomes
265:
library, removed a leaf of it (which is now in the
Cambridge University Library). In 1899 the Greek scholar Papadopoulos-Kerameus produced a catalog of the library's manuscripts and included a transcription of several lines of the partially visible underlying text. Upon seeing these lines
204:
A copy of
Isidorus' edition of Archimedes was made around AD 950 by an anonymous scribe, again in the Byzantine Empire, in a period during which the study of Archimedes flourished in Constantinople in a school founded by the mathematician, engineer, and former Greek Orthodox archbishop of
662:
the position of its center of mass might be restricted to lie at a certain distance from the fulcrum by a geometrical argument, by symmetry. The condition that the two figures balance now allows him to calculate the total mass of the other figure. He considered this method as a useful
800:
730:, he gives upper and lower bounds for the surface area of a sphere by cutting the sphere into sections of equal width. He then bounds the area of each section by the area of an inscribed and circumscribed cone, which he proves have a larger and smaller area correspondingly.
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When calculating approximating sums, he imposed the further constraint that the sums provide rigorous upper and lower bounds. This was required because the Greeks lacked algebraic methods that could establish that error terms in an approximation are
654:. He compared the area or volume of a figure of which he knew the total mass and center of mass with the area or volume of another figure he did not know anything about. He viewed plane figures as made out of infinitely many lines as in the later
371:
while in
Sirieix's cellar). This was directed by Dr. Will Noel, curator of manuscripts at the Walters Art Museum, and managed by Michael B. Toth of R.B. Toth Associates, with Dr. Abigail Quandt performing the conservation of the manuscript.
838:
to solve the puzzle, that is, to put the pieces back into their box. No pieces have been identified as such; the rules for placement, such as whether pieces are allowed to be turned over, are not known; and there is doubt about the board.
232:
of 177 later numbered leaves, of which 174 are extant (each older folded leaf became two leaves of the liturgical book). The palimpsest remained near
Jerusalem through at least the 16th century at the isolated Greek Orthodox monastery of
338:
grounds, and the palimpsest was bought for $ 2 million by an anonymous
American buyer. The lawyer who represented the anonymous buyer stated that the buyer was "a private American" who worked in "the high-tech industry", but was not
481:
On October 29, 2008 (the tenth anniversary of the purchase of the palimpsest at auction), all data, including images and transcriptions, were hosted on the Digital Palimpsest Web Page for free use under a
1157:
23:
A typical page from the Archimedes Palimpsest. The text of the prayer book is seen from top to bottom, the original Archimedes manuscript is seen as fainter text below it running from left to right
1668:
114:, possibly to protect it from occupying crusaders, who often equated Greek script with heresy against their Latin church and either burned or looted many such texts (including two additional
289:'s library (the Metochion of the Holy Sepulchre) in Constantinople in 1920. Shortly thereafter, during a turbulent period for the Greek community in Turkey that saw a Turkish victory in the
401:
religious images in the manuscript in an effort to increase its sales value. It appeared that these had rendered the underlying text forever illegible. However, in May 2005, highly focused
375:
The target audiences for the digitisation are Greek scholars, math historians, people building applications, libraries, archives, and scientists interested in the production of the images.
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remained unused by the author of the palimpsest and thus they are still lost. Between them, an announced result concerned the volume of the intersection of two cylinders, a figure that
409:
in Menlo Park, California, were used by Drs. Uwe Bergmann and Bob Morton to begin deciphering the parts of the 174-page text that had not yet been revealed. The production of X-ray
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to sell it in a public auction, risking an ownership dispute. The ownership of the palimpsest was immediately contested in federal court in New York in the case of the
224:
in 1204. There, in 1229, the Archimedes codex was unbound, scraped and washed, along with at least six other partial parchment manuscripts, including one with works of
635:
with rare brilliance. Later writers often criticized Archimedes for not explaining how he arrived at his results in the first place. This explanation is contained in
312:
Sirieix died in 1956, and in 1970 his daughter began attempting quietly to sell the valuable manuscript. Unable to sell it privately, in 1998 she finally turned to
1149:
330:. The Greek church contended that the palimpsest had been stolen from its library in Constantinople in the 1920s during a period of extreme persecution. Judge
965:
1968:
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in the Archimedes Palimpsest (shown after Suter from a different source; this version must be stretched to twice the width to conform to the Palimpsest)
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Either Archimedes used the Suter board, the pieces of which were allowed to be turned over, or the statistics of the Suter board are irrelevant.
447:
You start thinking striking one palimpsest is gold, and striking two is utterly astonishing. But then something even more extraordinary happened.
1000:
618:, an ancient Greek counterpart of the modern method of limits. Since the Greeks were aware that some numbers were irrational, their notion of a
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to the three color bands (red, green, and blue) of fluorescent light generated by ultraviolet illumination. Dr. Will Noel said in an interview:
3072:
1630:
298:
1228:
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367:, the palimpsest was the subject of an extensive imaging study from 1999 to 2008, and conservation (as it had suffered considerably from
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1989:
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486:, and processed images of the palimpsest in original page order were posted as a Google Book. In 2011, it was the subject of the
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and Nigel Wilson have produced a diplomatic transcription of the text, filling in gaps in Heiberg's account with these images.
666:
but always made sure to prove the results he found using exhaustion, since the method did not provide upper and lower bounds.
134:). The texts under the forged pictures, as well as previously unreadable texts, were revealed by analyzing images produced by
87:, sometime around AD 530. The copy found in the palimpsest was created from this original, also in Constantinople, during the
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guidelines, and metadata for the images and transcriptions included identification and cataloging information based on
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1584:, Oxford Studies in Ancient Philosophy 44:129–194 (2013); Porphyry is the preferred attribution see pp. 134, 137.
750:
is the calculation of the volume of a cylindrical wedge, a result that reappears as theorem XVII (schema XIX) of
319:
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All images and transcriptions are now freely available on the web at the Archimedes Digital Palimpsest under the
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The Greek Orthodox Patriarchate of Jerusalem v. Christies's Inc., 1999 U.S. Dist. LEXIS 13257 (S.D. N.Y. 1999)
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politician from the fourth century BC, which has also been found within the palimpsest. It is from his speech
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immediate. Rather, the first proposition sets up a board consisting of two squares side by side (as in
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467:, vol. 165, becoming the first new text from the palimpsest to be published in a scholarly journal.
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In April 2007, it was announced that a new text had been found in the palimpsest, a commentary on
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of matter—in this case, the previously hidden work of one of the founding fathers of all science.
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to solve problems about areas, volumes, and centers of gravity. Less attention was given to the
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and other authors. It contains two works of Archimedes that were thought to have been lost (the
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in the 1840s, and, intrigued by the Greek mathematics visible on the palimpsest he found in a
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But there are two essential differences between Archimedes' method and 19th-century methods:
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running to some 9 000 words. Most of this text was recovered in early 2009 by applying
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After imaging a page from the palimpsest, the original Archimedes text is now seen clearly
8:
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In his other works, Archimedes often proves the equality of two areas or volumes with
282:. Before that it was not widely known among mathematicians, physicists or historians.
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826:, a problem treated in the palimpsest that appears to deal with a children's puzzle.
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91:(c. AD 950), a time when mathematics in the capital was being revived by the former
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angles, the presence of right triangles makes the first proposition of Archimedes'
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arbitrarily fine. These proofs, still considered to be rigorous and correct, used
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925:
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198:
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This data is released for use under a Creative Commons license, with attribution
669:
Using this method, Archimedes was able to solve several problems now treated by
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262:
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84:
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1076:"X-Ray Fluorescence Imaging of the Archimedes Palimpsest: A Technical Summary"
855:). A reconciliation of the Suter board with this Codex board was published by
651:
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2419:
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1241:
871:
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325:
313:
1946:
1424:"Placed under X-ray gaze, Archimedes manuscript yields secrets lost to time"
110:
by Western crusaders in 1204, the manuscript was taken to an isolated Greek
75:. The first version of the compilation is believed to have been produced by
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817:
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In Heiberg's time, much attention was paid to Archimedes' brilliant use of
674:
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383:
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143:
80:
1150:"Revolutionary? Authentic? Stolen? The Story of the Archimedes Manuscript"
642:
The method that Archimedes describes was based upon his investigations of
301:, the palimpsest disappeared from the Greek church's library in Istanbul.
181:
Archimedes lived in the 3rd century BC and wrote his proofs as letters in
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A team of imaging scientists including Dr. Roger L. Easton, Jr. from the
279:
182:
135:
127:
59:
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Elements. The metadata and data were managed by Doug Emery of Emery IT.
216:
This medieval Byzantine manuscript then traveled from Constantinople to
163:
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2413:
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2287:
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2217:
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2157:
2097:
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2062:
2032:
1997:
1768:
1718:
694:
559:
517:
452:
340:
331:
228:. Their leaves were folded in half, rebound and reused for a Christian
225:
131:
54:
46:
1681:
2247:
2232:
2182:
2077:
2067:
2052:
2022:
1408:, Vol. 1. Catalogue and Commentary; Vol. 2. Images and Transcriptions
989:, Vol. 1. Catalogue and Commentary, Cambridge University press, 2011.
663:
565:
430:
242:
217:
40:
1501:. The Stoa Consortium for Electronic Publication in the Humanities.
27:
2384:
2162:
2007:
1494:
876:
698:
632:
492:
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237:. At some point before 1840 the palimpsest was brought back by the
234:
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111:
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2142:
852:
690:
643:
189:. These letters were first compiled into a comprehensive text by
19:
842:
The board illustrated here, as also by Netz, is one proposed by
673:, which was given its modern form in the seventeenth century by
470:
The transcriptions of the book were digitally encoded using the
2252:
2127:
1897:
713:
686:
781: = 4 Archimedean dome), whose volume relates to the
351:
2628:
2606:
2262:
1687:
1112:
880:
799:
157:
147:
43:
1314:"NOVA – Official Website – Inside the Archimedes Palimpsest"
69:) and the only surviving original Greek edition of his work
1580:
R. Chiaradonna, M. Rashed, D. Sedley and N. Tchernetska,
1318:
31:
Discovery reported in the New York Times on July 16, 1907
1371:"Archimedes Palimpsest Digital Release README Document"
197:
patriarchal church, sometime around AD 530 in the then
185:
addressed to contemporaries, including scholars at the
1039:
World Encyclopedia of Library and Information Services
451:
This referred to the previous discovery of a text by
681:. Among those problems were that of calculating the
592:
463:, and was published in 2008 in the academic journal
1404:Netz, Reviel; Noel, William; Wilson, Nigel (eds.).
1652:English translation (Heiberg's 1909 transcription)
3228:
1552:"Glassed-in DNA makes the ultimate time capsule"
334:decided in favor of Christie's Auction House on
603:The palimpsest contains the only known copy of
1976:
1962:
1703:
299:population exchange between Greece and Turkey
79:, the architect of the geometrically complex
1597:
1495:"The Digital Archimedes Palimpsest Released"
1411:(2011 ed.). Cambridge University Press.
1369:Toth, Mike; Emery, Doug (October 29, 2008).
985:Reviel Netz, William Noel and Nigel Wilson.
901:List of most expensive books and manuscripts
1177:
1175:
712:When rigorously proving theorems involving
346:
1969:
1955:
1710:
1696:
1462:Zeitschrift für Papyrologie und Epigraphik
1421:
1403:
1115:"The History of the Archimedes Manuscript"
738:to change variables or integrate by parts.
558:Speeches by the 4th-century BC politician
465:Zeitschrift für Papyrologie und Epigraphik
397:Sometime after 1938, a forger placed four
245:of the Holy Sepulcher) in Constantinople.
1665:(YouTube), Ignite (O'Reilly), August 2009
1368:
1181:
956:
954:
834:has argued that Archimedes discussed the
503:
222:Crusader sack of Byzantine Constantinople
1598:Dijksterhuis, E.J. Dijksterhuis (1987).
1172:
1143:
1141:
1139:
1137:
1135:
798:
697:, and the area of a region bounded by a
350:
287:Greek Orthodox Patriarchate of Jerusalem
239:Greek Orthodox Patriarchate of Jerusalem
162:
26:
18:
1442:
1345:"Archimedes Palimpsest – Press Release"
1285:
924:
3229:
2998:Latin translations of the 12th century
1516:
1492:
1147:
998:
951:
309:would later be required to reveal it.
2728:Straightedge and compass construction
1950:
1877:List of things named after Archimedes
1691:
1591:
1385:from the original on January 19, 2022
1132:
1108:
1106:
1104:
1102:
940:from the original on 19 February 2009
920:
918:
916:
2693:Incircle and excircles of a triangle
1582:A rediscovered Categories commentary
1549:
1422:Rock Woods, Heather (May 19, 2005).
1415:
1379:University of Pennsylvania Libraries
1256:
1225:University of Pennsylvania Libraries
1188:. Walker & Co, NY. p. 187.
1073:
746:A problem solved exclusively in the
1663:Restoring The Archimedes Palimpsest
1113:The Archimedes Palimpsest Project.
930:"Text Reveals More Ancient Secrets"
13:
3252:Scientific illuminated manuscripts
1643:The Nova Program teacher's version
1562:from the original on July 11, 2015
1240:All materials on OPenn are in the
1160:from the original on April 2, 2015
1148:Schulz, Matthias (June 22, 2007).
1099:
968:from the original on 8 August 2007
913:
879:gets the slave boy to consider in
407:Stanford Linear Accelerator Center
307:Stanford Linear Accelerator Center
14:
3273:
1797:The Method of Mechanical Theorems
1682:The Digital Archimedes Palimpsest
1675:
1634:, Weidenfeld & Nicolson, 2007
1550:Aron, Jacob (February 11, 2015).
1448:
1430:from the original on May 17, 2008
775:n = 4 Archimedean globe
707:Archimedes' use of infinitesimals
605:The Method of Mechanical Theorems
599:The Method of Mechanical Theorems
593:The Method of Mechanical Theorems
546:The Method of Mechanical Theorems
380:Rochester Institute of Technology
3237:Ancient Greek mathematical works
3210:
3197:
1931:
1930:
1814:
1493:Porter, Dot (October 29, 2008).
285:The manuscript was still in the
201:capital city of Constantinople.
16:Greek parchment codex manuscript
1574:
1543:
1532:from the original on 2023-04-22
1505:from the original on 2013-12-30
1486:
1475:from the original on 2011-11-18
1455:from the Archimedes Palimpsest"
1397:
1362:
1351:from the original on 2015-09-24
1337:
1326:from the original on 2017-08-15
1306:
1231:from the original on 2019-02-07
1213:
1202:from the original on 2023-04-22
1121:from the original on 2018-10-01
1088:from the original on 2017-05-18
1056:from the original on 2023-04-22
1019:from the original on 2023-04-22
3030:A History of Greek Mathematics
2543:The Quadrature of the Parabola
1717:
1628:Reviel Netz and William Noel.
1067:
1030:
992:
979:
962:"Editions of Archimedes' Work"
122:, attracting the attention of
1:
788:
705:. (For explicit details, see
689:, the center of gravity of a
66:Method of Mechanical Theorems
2811:Intersecting secants theorem
1748:On the Equilibrium of Planes
1264:"Reading Between the Lines,
964:. Brown University Library.
863:in March, 1926, sparking an
716:, Archimedes used a form of
511:On the Equilibrium of Planes
441:principal component analysis
220:, likely sometime after the
116:copies of Archimedes writing
7:
2806:Intersecting chords theorem
2673:Doctrine of proportionality
1836:Archimedes's cattle problem
1221:"The Archimedes Palimpsest"
894:
498:
291:Greco-Turkish War (1919–22)
187:Great Library of Alexandria
10:
3278:
2502:On the Sphere and Cylinder
2455:On the Sizes and Distances
1824:Discoveries and inventions
1762:On the Sphere and Cylinder
1755:Quadrature of the Parabola
1608:Princeton University Press
792:
727:On the Sphere and Cylinder
596:
532:On the Sphere and Cylinder
255:Constantin von Tischendorf
171:
3204:Ancient Greece portal
3193:
3143:
3021:
3008:Philosophy of mathematics
2978:
2971:
2945:
2923:Ptolemy's table of chords
2867:
2849:
2748:
2741:
2597:
2559:
2376:
1984:
1978:Ancient Greek mathematics
1926:
1890:
1859:
1823:
1812:
1725:
1657:Did Isaac Barrow read it?
1638:The Nova Program outlined
1539:– via Google Books.
1451:"Fragments of Hyperides'
1406:The Archimedes Palimpsest
1246:Creative Commons licenses
987:The Archimedes Palimpsest
320:Patriarchate of Jerusalem
274:text was translated into
248:
53:copy of a compilation of
3262:10th-century manuscripts
2875:Aristarchus's inequality
2448:On Conoids and Spheroids
1776:On Conoids and Spheroids
1375:archimedespalimpsest.net
1182:Hirshfeld, Alan (2009).
906:
580:Alexander of Aphrodisias
484:Creative Commons License
472:Text Encoding Initiative
347:Imaging and digitization
176:
155:Creative Commons License
2983:Ancient Greek astronomy
2796:Inscribed angle theorem
2786:Greek geometric algebra
2441:Measurement of a Circle
1734:Measurement of a Circle
1449:Carey, C.; et al.
999:Murray, Stuart (2009).
525:Measurement of a Circle
193:, the architect of the
167:Photo of the palimpsest
95:bishop of Thessaloniki
3217:Mathematics portal
3003:Non-Euclidean geometry
2958:Mouseion of Alexandria
2831:Tangent-secant theorem
2781:Geometric mean theorem
2766:Exterior angle theorem
2761:Angle bisector theorem
2465:On Sizes and Distances
1460:. Inhaltsverzeichnis.
1293:"archimedespalimpsest"
813:
656:method of indivisibles
504:Works contained within
449:
427:
356:
168:
112:monastery in Palestine
108:sack of Constantinople
89:Macedonian Renaissance
32:
24:
3242:Byzantine manuscripts
2905:Pappus's area theorem
2841:Theorem of the gnomon
2718:Quadratrix of Hippias
2641:Circles of Apollonius
2589:Problem of Apollonius
2567:Constructible numbers
2391:Archimedes Palimpsest
1872:Archimedes Palimpsest
1841:Archimedes' principle
1525:Archimedes Palimpsest
1299:on 21 February 2009.
802:
777:(and the half of it,
718:Cavalieri's principle
445:
419:
354:
253:The Biblical scholar
166:
120:Papadopoulos-Kerameus
37:Archimedes Palimpsest
30:
22:
3121:prehistoric counting
2918:Ptolemy's inequality
2859:Apollonius's theorem
2698:Method of exhaustion
2668:Diophantine equation
2658:Circumscribed circle
2475:On the Moving Sphere
1867:Archimedes' heat ray
1631:The Archimedes Codex
1266:Smithsonian Magazine
857:Richard Dixon Oldham
785:-polygonal pyramid.
616:method of exhaustion
476:Dublin Core Metadata
241:to its library (the
3257:Works by Archimedes
3207: •
3013:Neusis construction
2933:Spiral of Theodorus
2826:Pythagorean theorem
2771:Euclidean algorithm
2713:Lune of Hippocrates
2582:Squaring the circle
2338:Theon of Alexandria
2013:Aristaeus the Elder
1918:Eutocius of Ascalon
1908:Apollonius of Perga
1684:(official web site)
1250:Free Cultural Works
832:Stanford University
417:, director of SSRL:
392:Stanford University
191:Isidorus of Miletus
2900:Menelaus's theorem
2890:Irrational numbers
2703:Parallel postulate
2678:Euclidean geometry
2646:Apollonian circles
2188:Isidore of Miletus
1913:Hero of Alexandria
1851:Claw of Archimedes
1846:Archimedes's screw
1783:On Floating Bodies
1592:Additional sources
1244:or released under
814:
761:Some pages of the
539:On Floating Bodies
488:Walters Art Museum
361:Walters Art Museum
357:
209:, a cousin to the
169:
99:, a cousin of the
77:Isidore of Miletus
72:On Floating Bodies
33:
25:
3224:
3223:
3189:
3188:
2941:
2940:
2928:Ptolemy's theorem
2801:Intercept theorem
2651:Apollonian gasket
2577:Doubling the cube
2550:The Sand Reckoner
1944:
1943:
1903:Eudoxus of Cnidus
1882:Pseudo-Archimedes
1831:Archimedean solid
1741:The Sand Reckoner
1671:(via Archive.org)
1606:. Princeton, NJ:
867:craze that year.
810:dissection puzzle
683:center of gravity
679:Gottfried Leibniz
671:integral calculus
422:Synchrotron light
413:was described by
3269:
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3214:
3202:
3201:
3200:
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2975:
2963:Platonic Academy
2910:Problem II.8 of
2880:Crossbar theorem
2836:Thales's theorem
2776:Euclid's theorem
2746:
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2663:Commensurability
2624:Axiomatic system
2572:Angle trisection
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564:A commentary on
405:produced at the
207:Leo the Geometer
97:Leo the Geometer
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2733:Triangle center
2723:Regular polygon
2600:and definitions
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399:Byzantine-style
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318:Greek Orthodox
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293:along with the
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836:number of ways
793:Main article:
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648:center of mass
627:, by U and L.
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3247:Palimpsests
3104:mathematics
2912:Arithmetica
2509:Ostomachion
2478:(Autolycus)
2397:Arithmetica
2173:Hippocrates
2103:Dinostratus
2088:Dicaearchus
2018:Aristarchus
1790:Ostomachion
1661:Will Noel:
1610:. pp.
1434:February 8,
1154:Der Spiegel
1002:The Library
865:Ostomachion
849:Ostomachion
828:Reviel Netz
823:Ostomachion
805:Ostomachion
795:Ostomachion
685:of a solid
620:real number
585:Other works
553:Ostomachion
493:integration
388:Reviel Netz
280:T. L. Heath
183:Doric Greek
136:ultraviolet
128:World War I
60:Ostomachion
3231:Categories
3156:Babylonian
3056:arithmetic
3022:History of
2851:Apollonius
2536:(Menelaus)
2495:On Spirals
2414:Catoptrics
2353:Xenocrates
2348:Thymaridas
2333:Theodosius
2318:Theaetetus
2298:Simplicius
2288:Pythagoras
2273:Posidonius
2258:Philonides
2218:Nicomachus
2213:Metrodorus
2203:Menaechmus
2158:Hipparchus
2148:Heliodorus
2098:Diophantus
2083:Democritus
2063:Chrysippus
2033:Archimedes
2028:Apollonius
1998:Anaxagoras
1990:(timeline)
1769:On Spirals
1719:Archimedes
1648:The Method
1602:Archimedes
1566:January 4,
1536:2009-03-31
1509:2013-12-29
1479:2009-10-11
1389:January 4,
1355:2015-03-16
1330:2017-08-24
1278:2009-03-31
1235:2016-08-01
1206:2013-09-29
1185:Eureka Man
1125:2015-03-16
1092:2013-09-29
1060:2020-10-29
1023:2020-10-29
972:2007-07-23
944:2009-03-31
859:, FRS, in
789:Stomachion
695:paraboloid
687:hemisphere
637:The Method
623:found, or
571:Categories
560:Hypereides
518:On Spirals
453:Hypereides
436:Categories
341:Bill Gates
332:Kimba Wood
326:Christie's
314:Christie's
226:Hypereides
132:Jeff Bezos
55:Archimedes
47:palimpsest
2617:Inscribed
2377:Treatises
2368:Zenodorus
2328:Theodorus
2303:Sosigenes
2248:Philolaus
2233:Oenopides
2228:Nicoteles
2223:Nicomedes
2183:Hypsicles
2078:Ctesibius
2068:Cleomedes
2053:Callippus
2038:Autolycus
2023:Aristotle
2003:Anthemius
1164:March 16,
664:heuristic
646:, on the
625:exhausted
566:Aristotle
431:Aristotle
365:Baltimore
243:Metochion
218:Jerusalem
211:patriarch
101:Patriarch
41:parchment
3181:Japanese
3166:Egyptian
3109:timeline
3097:timeline
3085:timeline
3080:geometry
3073:timeline
3068:calculus
3061:timeline
3049:timeline
2752:Elements
2598:Concepts
2560:Problems
2533:Spherics
2523:Spherics
2488:(Euclid)
2434:(Euclid)
2431:Elements
2424:(Euclid)
2385:Almagest
2293:Serenus
2268:Porphyry
2208:Menelaus
2163:Hippasus
2138:Eutocius
2113:Domninus
2008:Archytas
1936:Category
1560:Archived
1530:Archived
1503:Archived
1499:stoa.org
1470:Archived
1468:: 1–19.
1428:Archived
1383:Archived
1349:Archived
1324:Archived
1229:Archived
1200:Archived
1158:Archived
1119:Archived
1083:Archived
1054:Archived
1042:. 1993.
1017:Archived
966:Archived
938:Archived
934:BBC News
895:See also
877:Socrates
699:parabola
650:and the
633:geometry
589:Source:
576:Porphyry
499:Contents
457:Athenian
257:visited
235:Mar Saba
140:infrared
63:and the
3161:Chinese
3116:numbers
3044:algebra
2972:Related
2946:Centers
2742:Results
2612:Central
2283:Ptolemy
2278:Proclus
2243:Perseus
2198:Marinus
2178:Hypatia
2168:Hippias
2143:Geminus
2133:Eudoxus
2123:Eudemus
2093:Diocles
870:Modern
853:Tangram
767:Apostol
691:frustum
644:physics
612:Eudoxus
578:(or by
359:At the
276:English
172:History
3176:Indian
2953:Cyrene
2485:Optics
2404:Conics
2323:Theano
2313:Thales
2308:Sporus
2253:Philon
2238:Pappus
2128:Euclid
2058:Carpus
2048:Bryson
1898:Euclid
1618:
1614:–133.
1192:
1046:
1009:
861:Nature
763:Method
752:Kepler
748:Method
742:small.
714:volume
403:X-rays
336:laches
249:Modern
146:, and
3171:Incan
3092:logic
2868:Other
2636:Chord
2629:Axiom
2607:Angle
2263:Plato
2153:Heron
2073:Conon
1473:(PDF)
1458:(PDF)
1086:(PDF)
1079:(PDF)
907:Notes
881:Plato
808:is a
724:. In
455:, an
328:, Inc
272:Greek
177:Early
158:CC BY
148:X-ray
44:codex
39:is a
3133:list
2421:Data
2193:Leon
2043:Bion
1616:ISBN
1568:2022
1436:2016
1391:2022
1190:ISBN
1166:2015
1044:ISBN
1007:ISBN
885:Meno
769:and
677:and
369:mold
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